https://file.aiquickdraw.com/mathbot/file/1728312332850mg9hm6tb.jpg

https://file.aiquickdraw.com/mathbot/file/1728312332851ixd6ash4.jpg

For the two graphs you've provided, the question asks whether each represents a function. Let's analyze them based on the definition of a function:

1. **Question 9 (First graph)**:
   - A relation is a function if each input (on the x-axis) has exactly one corresponding output (on the y-axis). This means, for every \( x \)-value (Speed in mph), there should be only one \( y \)-value (Distance from home).
   - In the first graph, some vertical lines (especially around 60 mph) would intersect the curve at multiple points. This indicates that a single \( x \)-value corresponds to multiple \( y \)-values, violating the definition of a function.
   - **Answer**: No, this does not represent a function.

2. **Question 10 (Second graph)**:
   - This is a scatter plot, where each point corresponds to a pair of values (Years since 2000, Gross Sales in millions). For it to represent a function, each input \( x \)-value (Years since 2000) must correspond to exactly one \( y \)-value (Gross Sales).
   - Here, each \( x \)-value seems to have only one corresponding \( y \)-value (no repeated years with different sales values), so this is indeed a function.
   - **Answer**: Yes, this represents a function.

Would you like further details on how to interpret graphs like these, or any additional help? 

Here are 5 related questions to expand on this topic:

1. What is the vertical line test and how does it help identify functions?
2. How would the interpretation change if there were repeated points in the second graph?
3. Can a curve be a function even if it changes direction multiple times?
4. How do piecewise functions relate to these types of graphs?
5. What are some real-world examples where graphs fail to represent functions?

**Tip**: When trying to determine if a relation is a function, remember that a function cannot have two different outputs for the same input!

Does the graph in question represent a function?

Analyze two graphs to determine if they represent functions based on the vertical line test. Explore key concepts such as functions, relations, and how to interpret graphs in mathematics. Suitable for students in Grades 8-10.

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